REAL

Squares and their centers

Keleti, Tamás and Nagy, Dániel and Shmerkin, P. (2018) Squares and their centers. JOURNAL D ANALYSE MATHEMATIQUE, 134 (2). pp. 643-669. ISSN 0021-7670

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Abstract

We study the relationship between the size of two sets B, S ⊂ R2, when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]2, prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others. © 2018, Hebrew University Magnes Press.

Item Type: Article
Uncontrolled Keywords: PLANE; Circles; MAXIMAL OPERATORS;
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 12 Jan 2019 15:08
Last Modified: 12 Jan 2019 15:08
URI: http://real.mtak.hu/id/eprint/89807

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